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<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">Fitting point-spread function models</h1>

<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Least-squares methods</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<p class="ltx_p">To approximate the data with a point-spread function, least-squares
methods <cite class="ltx_cite">[<a href="#bib.bib19" title="The Advanced Theory of Statistics" class="ltx_ref">4</a>, <a href="#bib.bib4" title="Data reduction and error analysis for the physical science" class="ltx_ref">1</a>, <a href="#bib.bib23" title="Optimized localization analysis for single-molecule tracking and super-resolution microscopy" class="ltx_ref">5</a>]</cite> are employed
to minimize the sum of (weighted) squared residuals defined by
</p>
<table id="Sx1.E1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E1.m1" class="ltx_Math" style="vertical-align:-27px" src="mi/mi63.png" width="375" height="53" alt="\chi^{2}\left(\boldsymbol{\theta}\mid\mathcal{D}\right)=\sum_{x,y\in\mathcal{D%
}}{w\left(\tilde{I}\left(x,y\right)-\mathrm{PSF}\left(x,y\mid\boldsymbol{%
\theta}\right)\right)^{2}}\,."></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(1)</span></td>
</tr>
</table>
<p class="ltx_p">Here the residual value for the <img id="Sx1.SSx1.p1.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi48.png" width="45" height="21" alt="\left(x,y\right)"> data point is
defined as the difference between the observed image intensity <img id="Sx1.SSx1.p1.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi79.png" width="57" height="24" alt="\tilde{I}\left(x,y\right)">
and the value approximated by the <img id="Sx1.SSx1.p1.m3" class="ltx_Math" style="vertical-align:-6px" src="mi/mi76.png" width="106" height="21" alt="\mathrm{PSF}\left(x,y\mid\boldsymbol{\theta}\right)">,
where <img id="Sx1.SSx1.p1.m4" class="ltx_Math" style="vertical-align:-2px" src="mi/mi47.png" width="15" height="16" alt="\boldsymbol{\theta}"> are the PSF parameters. The residual
value can be further weighted by <img id="Sx1.SSx1.p1.m5" class="ltx_Math" style="vertical-align:-2px" src="mi/mi81.png" width="49" height="16" alt="w=1">, making all measurements equally
significant, or weighted by <img id="Sx1.SSx1.p1.m6" class="ltx_Math" style="vertical-align:-6px" src="mi/mi80.png" width="111" height="24" alt="w=1/\tilde{I}\left(x,y\right)">, which
takes into account the uncertainty in the number of detected photons.</p>
</div>
<div id="Sx1.SSx1.p2" class="ltx_para">
<p class="ltx_p">The search for parameters <img id="Sx1.SSx1.p2.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi67.png" width="15" height="20" alt="\boldsymbol{\hat{\theta}}"> which minimize
<img id="Sx1.SSx1.p2.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi69.png" width="79" height="23" alt="\chi^{2}\left(\boldsymbol{\theta}\mid\mathcal{D}\right)">, leads
to an optimization problem formulated as</p>
<table id="Sx1.E2" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E2.m1" class="ltx_Math" style="vertical-align:-19px" src="mi/mi62.png" width="183" height="37" alt="\boldsymbol{\hat{\theta}}=\argmin_{\boldsymbol{\theta}}\chi^{2}\left(%
\boldsymbol{\theta}\mid\mathcal{D}\right)\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(2)</span></td>
</tr>
</table>
<p class="ltx_p">which we solve by the Levenberg-Marquardt algorithm as
implemented in the Apache Commons Math library <cite class="ltx_cite">[<a href="#bib.bib6" title="The Apache Commons Mathematics Library; version 3.2" class="ltx_ref">2</a>]</cite>.
The starting point for the optimization process is computed from the
data as the difference between the maximum and the minimum intensity
values for the molecular intensity <img id="Sx1.SSx1.p2.m3" class="ltx_Math" style="vertical-align:-5px" src="mi/mi52.png" width="26" height="19" alt="\theta_{N}">, and as the minimum
intensity value for the background offset <img id="Sx1.SSx1.p2.m4" class="ltx_Math" style="vertical-align:-5px" src="mi/mi53.png" width="20" height="19" alt="\theta_{b}">. Users have
to choose the starting point for the approximate molecular width <img id="Sx1.SSx1.p2.m5" class="ltx_Math" style="vertical-align:-5px" src="mi/mi78.png" width="22" height="19" alt="\theta_{\sigma}">.
The sub-pixel refinement of the coordinates is obtained as <img id="Sx1.SSx1.p2.m6" class="ltx_Math" style="vertical-align:-5px" src="mi/mi70.png" width="63" height="23" alt="\hat{x}_{0}=\hat{\theta}_{x}">
and <img id="Sx1.SSx1.p2.m7" class="ltx_Math" style="vertical-align:-7px" src="mi/mi71.png" width="61" height="26" alt="\hat{y}_{0}=\hat{\theta}_{y}">, where <img id="Sx1.SSx1.p2.m8" class="ltx_Math" style="vertical-align:-14px" src="mi/mi66.png" width="125" height="35" alt="\boldsymbol{\hat{\theta}}=\left[\hat{\theta}_{x},\hat{\theta}_{y},\ldots\right]">.</p>
</div>
</div>
<div id="Sx1.SSx2" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Maximum-likelihood estimation</h2>

<div id="Sx1.SSx2.p1" class="ltx_para">
<p class="ltx_p">This approach assumes that the number of photons collected by a single
camera pixel follows the Poisson distribution. Thus, the probability
of <img id="Sx1.SSx2.p1.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi73.png" width="14" height="12" alt="\kappa"> photons arriving at a camera pixel, where the expected
number of photons is <img id="Sx1.SSx2.p1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi75.png" width="14" height="16" alt="\lambda">, is given by</p>
<table id="Sx1.E3" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E3.m1" class="ltx_Math" style="vertical-align:-14px" src="mi/mi64.png" width="147" height="43" alt="p\left(\kappa\mid\lambda\right)=\frac{\lambda^{\kappa}\mathrm{e}^{-\lambda}}{%
\kappa!}\,."></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(3)</span></td>
</tr>
</table>
</div>
<div id="Sx1.SSx2.p2" class="ltx_para">
<p class="ltx_p">Suppose that samples are drawn independently from the Poisson distribution,
with the expected photon count <img id="Sx1.SSx2.p2.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi74.png" width="140" height="21" alt="\lambda=\mathrm{PSF}\left(x,y\mid\boldsymbol{\theta}\right)">
given by the point-spread function model, and the observed photon
count <img id="Sx1.SSx2.p2.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi72.png" width="91" height="24" alt="\kappa=\tilde{I}\left(x,y\right)"> given by the image intensity
expressed in photons. The likelihood <cite class="ltx_cite">[<a href="#bib.bib19" title="The Advanced Theory of Statistics" class="ltx_ref">4</a>, <a href="#bib.bib23" title="Optimized localization analysis for single-molecule tracking and super-resolution microscopy" class="ltx_ref">5</a>, <a href="#bib.bib33" title="Fast, single-molecule localization that achieves theoretically minimum uncertainty" class="ltx_ref">7</a>, <a href="#bib.bib15" title="Simultaneous multiple-emitter fitting for single molecule super-resolution imaging" class="ltx_ref">3</a>]</cite>
of the parameters <img id="Sx1.SSx2.p2.m3" class="ltx_Math" style="vertical-align:-2px" src="mi/mi47.png" width="15" height="16" alt="\boldsymbol{\theta}"> can be modeled as</p>
<table id="Sx1.E4" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E4.m1" class="ltx_Math" style="vertical-align:-27px" src="mi/mi60.png" width="380" height="61" alt="L(\boldsymbol{\theta}\mid\mathcal{D})=\prod_{x,y\in\mathcal{D}}\frac{\mathrm{%
PSF}\left(x,y\mid\boldsymbol{\theta}\right)^{\tilde{I}\left(x,y\right)}\mathrm%
{e}^{-\mathrm{PSF}\left(x,y\mid\boldsymbol{\theta}\right)}}{\tilde{I}\left(x,y%
\right)!}\,."></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(4)</span></td>
</tr>
</table>
</div>
<div id="Sx1.SSx2.p3" class="ltx_para">
<p class="ltx_p">The maximum likelihood estimate of the parameters <img id="Sx1.SSx2.p3.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi47.png" width="15" height="16" alt="\boldsymbol{\theta}">
is, by definition, the value that maximizes the likelihood <img id="Sx1.SSx2.p3.m2" class="ltx_Math" style="vertical-align:-6px" src="mi/mi65.png" width="69" height="21" alt="L(\boldsymbol{\theta}\mid\mathcal{D})">.
Intuitively, the estimate <img id="Sx1.SSx2.p3.m3" class="ltx_Math" style="vertical-align:-2px" src="mi/mi67.png" width="15" height="20" alt="\boldsymbol{\hat{\theta}}"> corresponds
to the value <img id="Sx1.SSx2.p3.m4" class="ltx_Math" style="vertical-align:-2px" src="mi/mi47.png" width="15" height="16" alt="\boldsymbol{\theta}"> that best agrees with the data.
The maximization problem has the form</p>
<table id="Sx1.E5" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E5.m1" class="ltx_Math" style="vertical-align:-27px" src="mi/mi61.png" width="482" height="49" alt="\boldsymbol{\hat{\theta}}=\argmax_{\boldsymbol{\theta}}\sum_{x,y\in\mathcal{D}%
}{\left[\tilde{I}\left(x,y\right)\ln{\left(\mathrm{PSF}\left(x,y\mid%
\boldsymbol{\theta}\right)\right)}-\mathrm{PSF}\left(x,y\mid\boldsymbol{\theta%
}\right)\right]}\,,"></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(5)</span></td>
</tr>
</table>
<p class="ltx_p">which we solve by the Nelder-Mead method <cite class="ltx_cite">[<a href="#bib.bib25" title="Algorithm AS 47–function minimization using a simplex procedure" class="ltx_ref">6</a>]</cite>. The starting
point for the optimization process is computed from the data as the
difference between the maximum and the minimum intensity values for
the molecular intensity <img id="Sx1.SSx2.p3.m5" class="ltx_Math" style="vertical-align:-5px" src="mi/mi52.png" width="26" height="19" alt="\theta_{N}">, and as the minimum intensity
value for the background offset <img id="Sx1.SSx2.p3.m6" class="ltx_Math" style="vertical-align:-5px" src="mi/mi53.png" width="20" height="19" alt="\theta_{b}">. Users have to choose
the starting point for the approximate molecular width <img id="Sx1.SSx2.p3.m7" class="ltx_Math" style="vertical-align:-5px" src="mi/mi78.png" width="22" height="19" alt="\theta_{\sigma}">.
The sub-pixel refinement of the coordinates is obtained as <img id="Sx1.SSx2.p3.m8" class="ltx_Math" style="vertical-align:-5px" src="mi/mi70.png" width="63" height="23" alt="\hat{x}_{0}=\hat{\theta}_{x}">
and <img id="Sx1.SSx2.p3.m9" class="ltx_Math" style="vertical-align:-7px" src="mi/mi71.png" width="61" height="26" alt="\hat{y}_{0}=\hat{\theta}_{y}">, where <img id="Sx1.SSx2.p3.m10" class="ltx_Math" style="vertical-align:-14px" src="mi/mi66.png" width="125" height="35" alt="\boldsymbol{\hat{\theta}}=\left[\hat{\theta}_{x},\hat{\theta}_{y},\ldots\right]">.</p>
</div>
</div>
<div id="Sx1.SSx3" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Constraining parameters of PSF models</h2>

<div id="Sx1.SSx3.p1" class="ltx_para">
<p class="ltx_p">The Levenberg-Marquardt algorithm and the Nelder-Mead method used
above search for values of the parameters <img id="Sx1.SSx3.p1.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi47.png" width="15" height="16" alt="\boldsymbol{\theta}"> over
an infinite interval. The optimization process can therefore converge
to a solution with negative values which is impossible for variables
corresponding to image intensity or to the standard deviation of a
Gaussian PSF. We therefore limit the interval of possible values by
transforming the relevant parameters and using <img id="Sx1.SSx3.p1.m2" class="ltx_Math" style="vertical-align:-14px" src="mi/mi77.png" width="114" height="35" alt="\mathrm{PSF}\left(x,y\mid\boldsymbol{\tilde{\theta}}\right)">
in Equations (<a href="#Sx1.E2" title="(2) ‣ Least-squares methods ‣ Fitting point-spread function models" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>) and (<a href="#Sx1.E5" title="(5) ‣ Maximum-likelihood estimation ‣ Fitting point-spread function models" class="ltx_ref"><span class="ltx_text ltx_ref_tag">5</span></a>) instead of <img id="Sx1.SSx3.p1.m3" class="ltx_Math" style="vertical-align:-6px" src="mi/mi76.png" width="106" height="21" alt="\mathrm{PSF}\left(x,y\mid\boldsymbol{\theta}\right)">.
The transformation for a 2D Gaussian PSF model is <img id="Sx1.SSx3.p1.m4" class="ltx_Math" style="vertical-align:-8px" src="mi/mi68.png" width="174" height="27" alt="\boldsymbol{\tilde{\theta}}=\left[\theta_{x},\theta_{y},\theta_{\sigma}^{2},%
\theta_{N}^{2},\theta_{b}^{2}\right]">.
The optimization process is still unconstrained but will result in
positive PSF parameters. Such a transformation also improves the
stability of the fit.</p>
</div>
</div>
<div id="Sx1.SSx4" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Guidelines for the choice of parameters</h2>

<div id="Sx1.SSx4.p1" class="ltx_para">
<p class="ltx_p">Fitting PSF models by maximum likelihood estimation generally gives
slightly better results than fitting by least square methods, particularly
when the photon counts are low. Maximum likelihood estimation requires
correct setup of the <a href="../../CameraSetupPlugIn.html" title="" class="ltx_ref">camera parameters</a>
(photoelectrons per A/D count and the digitizer offset level). The
recommended PSF model, which generally works well, is the <a href="IntSymmetricGaussianEstimatorUI.html" title="" class="ltx_ref">integrated Gaussian</a>.
The initial size of sigma can be found by running ThunderSTORM on
few images of the data sequence. A histogram of the fitted sizes of
sigma (in pixels) can help to find the initial value. The size of
the fitting radius should be an integer number close to 3*sigma.</p>
</div>
</div>
<div id="Sx1.SSx5" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx5.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="Estimators.html" title="" class="ltx_ref">Sub-pixel 2D localization of molecules</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="PSF.html" title="" class="ltx_ref">Point-spread function (PSF)</a></p>
</div>
</li>
<li id="I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i3.p1" class="ltx_para">
<p class="ltx_p"><a href="FittingRegion.html" title="" class="ltx_ref">Definition of the fitting region</a></p>
</div>
</li>
<li id="I1.i4" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i4.p1" class="ltx_para">
<p class="ltx_p"><a href="LocalizationUncertainty.html" title="" class="ltx_ref">Localization uncertainty</a></p>
</div>
</li>
<li id="I1.i5" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i5.p1" class="ltx_para">
<p class="ltx_p"><a href="CrowdedField.html" title="" class="ltx_ref">Multiple-emitter fitting analysis</a></p>
</div>
</li>
<li id="I1.i6" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i6.p1" class="ltx_para">
<p class="ltx_p"><a href="../../CameraSetupPlugIn.html" title="" class="ltx_ref">Camera setup</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib4" class="ltx_bibitem ltx_bib_book">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">P. R. Bevington and D. K. Robinson</span><span class="ltx_text ltx_bib_year">(2003)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Data reduction and error analysis for the physical science</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_series">McGraw-Hill Higher Education</span>,  <span class="ltx_text ltx_bib_publisher">McGraw-Hill</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><span class="ltx_text isbn ltx_bib_external">ISBN 9780072472271</span></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx1.p1" title="Least-squares methods ‣ Fitting point-spread function models" class="ltx_ref"><span class="ltx_text ltx_ref_title">Least-squares methods</span></a>.
</span>
</li>
<li id="bib.bib6" class="ltx_bibitem ltx_bib_misc">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[2]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">Commons-Math</span><span class="ltx_text ltx_bib_year">(2013-04)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">The Apache Commons Mathematics Library; version 3.2</span>,
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://commons.apache.org/proper/commons-math/" title="" class="ltx_ref ltx_bib_external">Link</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx1.p2" title="Least-squares methods ‣ Fitting point-spread function models" class="ltx_ref"><span class="ltx_text ltx_ref_title">Least-squares methods</span></a>.
</span>
</li>
<li id="bib.bib15" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[3]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">F. Huang, S. L. Schwartz, J. M. Byars and K. A. Lidke</span><span class="ltx_text ltx_bib_year">(2011)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Simultaneous multiple-emitter fitting for single molecule super-resolution imaging</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Biomedical Optics Express</span> <span class="ltx_text ltx_bib_volume">2</span> (<span class="ltx_text ltx_bib_number">5</span>), <span class="ltx_text ltx_bib_pages"> pp. 1377–93</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1364/BOE.2.001377" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx2.p2" title="Maximum-likelihood estimation ‣ Fitting point-spread function models" class="ltx_ref"><span class="ltx_text ltx_ref_title">Maximum-likelihood estimation</span></a>.
</span>
</li>
<li id="bib.bib19" class="ltx_bibitem ltx_bib_book">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[4]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">M. Kendall and A. Stuart</span><span class="ltx_text ltx_bib_year">(1979)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">The Advanced Theory of Statistics</span>,
</span>
<span class="ltx_bibblock"> <span class="ltx_text ltx_bib_publisher">London: Charles Griffin</span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx1.p1" title="Least-squares methods ‣ Fitting point-spread function models" class="ltx_ref"><span class="ltx_text ltx_ref_title">Least-squares methods</span></a>,
<a href="#Sx1.SSx2.p2" title="Maximum-likelihood estimation ‣ Fitting point-spread function models" class="ltx_ref"><span class="ltx_text ltx_ref_title">Maximum-likelihood estimation</span></a>.
</span>
</li>
<li id="bib.bib23" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[5]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">K. I. Mortensen, L. S. Churchman, J. A. Spudich and H. Flyvbjerg</span><span class="ltx_text ltx_bib_year">(2010)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Optimized localization analysis for single-molecule tracking and super-resolution microscopy</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Nature Methods</span> <span class="ltx_text ltx_bib_volume">7</span> (<span class="ltx_text ltx_bib_number">5</span>), <span class="ltx_text ltx_bib_pages"> pp. 377–381</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1038/nmeth.1447" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
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<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx1.p1" title="Least-squares methods ‣ Fitting point-spread function models" class="ltx_ref"><span class="ltx_text ltx_ref_title">Least-squares methods</span></a>,
<a href="#Sx1.SSx2.p2" title="Maximum-likelihood estimation ‣ Fitting point-spread function models" class="ltx_ref"><span class="ltx_text ltx_ref_title">Maximum-likelihood estimation</span></a>.
</span>
</li>
<li id="bib.bib25" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[6]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">R. O’Neill</span><span class="ltx_text ltx_bib_year">(1971)</span>
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<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Algorithm AS 47–function minimization using a simplex procedure</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Applied Statistics</span> <span class="ltx_text ltx_bib_volume">20</span>, <span class="ltx_text ltx_bib_pages"> pp. 338–45</span>.
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<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://www.jstor.org/stable/2346772" title="" class="ltx_ref ltx_bib_external">Link</a></span>.
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<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx2.p3" title="Maximum-likelihood estimation ‣ Fitting point-spread function models" class="ltx_ref"><span class="ltx_text ltx_ref_title">Maximum-likelihood estimation</span></a>.
</span>
</li>
<li id="bib.bib33" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[7]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">C. S. Smith, N. Joseph, B. Rieger and K. A. Lidke</span><span class="ltx_text ltx_bib_year">(2010)</span>
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<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Fast, single-molecule localization that achieves theoretically minimum uncertainty</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Nature Methods</span> <span class="ltx_text ltx_bib_volume">7</span> (<span class="ltx_text ltx_bib_number">5</span>), <span class="ltx_text ltx_bib_pages"> pp. 373–5</span>.
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<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1038/nmeth.1449" title="" class="ltx_ref doi ltx_bib_external">Document</a>,
<span class="ltx_text issn ltx_bib_external">ISSN 1548-7105</span></span>.
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<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx2.p2" title="Maximum-likelihood estimation ‣ Fitting point-spread function models" class="ltx_ref"><span class="ltx_text ltx_ref_title">Maximum-likelihood estimation</span></a>.
</span>
</li>
</ul>
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